A measurement of observables sensitive to spin correlations in $tt¯$ production is presented, using 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collision data at $s=13$ TeV recorded with the ATLAS detector at the Large Hadron Collider. Differential cross-sections are measured in events with exactly one electron and one muon with opposite-sign electric charge as a function of the azimuthal opening angle and the absolute difference in pseudorapidity between the electron and muon candidates in the laboratory frame. The azimuthal opening angle is also measured as a function of the invariant mass of the $tt¯$ system. The measured differential cross-sections are compared to predictions by several NLO Monte Carlo generators and fixed-order calculations. The observed degree of spin correlation is somewhat higher than predicted by the generators used. The data are consistent with the prediction of one of the fixed-order calculations at NLO, but agree less well with higher-order predictions. Using these leptonic observables, a search is performed for pair production of supersymmetric top squarks decaying into Standard Model top quarks and light neutralinos. Top squark masses between 170 and 230 GeV are largely excluded at the 95% confidence level for kinematically allowed values of the neutralino mass.